The Modern Mathematics of Deep Learning

A Fourier frequency bound

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 4.2 Approximating with random pieces

The numerator is the magnitude of a complex-number difference. The source uses this bound when x lies in the unit ball and ξ is nonzero.

In one dimension, cosine and sine give the real and imaginary parts.

This inequality controls the sampling-error expression in the source argument.

Source formula example

|e^{2\pi i\langle x,\xi\rangle}-1|/\|\xi\|_{2}\leq 2\pi

illustration · The code evaluates one point; it does not prove the inequality. The denominator requires ξ≠0.

Symbols

$x$: A real d-vector in the unit ball.
$ξ$: A nonzero real d-vector.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

xNonzero ξ

This example’s data flow

You choose

input.x = 0.5
input.xi = 1

Calculated here

Result expression: { real, imaginary, ratio, upperBound: 2 * Math.PI }

x = input.x Uses: input.x
xi = input.xi Uses: input.xi
angle = 2 * Math.PI * x * xi Uses: x, xi
real = Math.cos(angle) - 1 Uses: angle
imaginary = Math.sin(angle) Uses: angle
ratio = Math.hypot(real, imaginary) / Math.abs(xi) Uses: real, imaginary, xi

Returns

Example result

real: -2
imaginary: 0
ratio: 2
upperBound: ≈6.283

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The source inequality follows from |exp(it)−1|≤|t| and the Cauchy–Schwarz inequality. The toy is one-dimensional. The limiting expression at ξ=0 needs separate interpretation and is not divided by zero.

Source: Frequency inequality